### Abstract

Original language | English |
---|---|

Pages (from-to) | 323-333 |

Journal | Mathematical Proceedings of the Cambridge Philosophical Society |

Volume | 93 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1983 |

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*Mathematical Proceedings of the Cambridge Philosophical Society*,

*93*(2), 323-333. https://doi.org/10.1017/S030500410006062X

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*Mathematical Proceedings of the Cambridge Philosophical Society*, vol. 93, no. 2, pp. 323-333. https://doi.org/10.1017/S030500410006062X

**Wavefronts and parallels in Euclidean space.** / Bruce, J.W.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Wavefronts and parallels in Euclidean space

AU - Bruce, J.W.

PY - 1983

Y1 - 1983

N2 - Given a smooth plane curve or surface in 3 its parallels consist of those curves or surfaces a fixed distance d down the normals in a fixed direction. Generically they have Legendre singularities. We are concerned here with the way in which these parallels change as we alter the distance d. (Alternatively the manner in which wave-fronts change as they evolve from an initial smooth wavefront.) This problem was considered in (1) by V. I. Arnold. In a very beautiful paper he describes the generic evolution of wavefronts but does not prove that for a generic initial wavefront in 2 or 3 the evolution is of the type described there. This we do here, using the tool of transversality. A more positive outcome of our investigation is that some of Arnold's generic forms do not occur (those corresponding to A2 singularities).

AB - Given a smooth plane curve or surface in 3 its parallels consist of those curves or surfaces a fixed distance d down the normals in a fixed direction. Generically they have Legendre singularities. We are concerned here with the way in which these parallels change as we alter the distance d. (Alternatively the manner in which wave-fronts change as they evolve from an initial smooth wavefront.) This problem was considered in (1) by V. I. Arnold. In a very beautiful paper he describes the generic evolution of wavefronts but does not prove that for a generic initial wavefront in 2 or 3 the evolution is of the type described there. This we do here, using the tool of transversality. A more positive outcome of our investigation is that some of Arnold's generic forms do not occur (those corresponding to A2 singularities).

U2 - 10.1017/S030500410006062X

DO - 10.1017/S030500410006062X

M3 - Article

VL - 93

SP - 323

EP - 333

JO - Mathematical Proceedings of the Cambridge Philosophical Society

JF - Mathematical Proceedings of the Cambridge Philosophical Society

SN - 0305-0041

IS - 2

ER -